Theorem bd0 13193

Description: A formula equivalent to a bounded one is bounded. See also bd0r 13194. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0.min	⊢ BOUNDED 𝜑
bd0.maj	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bd0	⊢ BOUNDED 𝜓

Proof of Theorem bd0

Step	Hyp	Ref	Expression
1		bd0.min	. 2 ⊢ BOUNDED 𝜑
2		bd0.maj	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	ax-bd0 13182	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
4	1, 3	ax-mp 5	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 104  BOUNDED wbd 13181

This theorem was proved from axioms:  ax-mp 5  ax-bd0 13182

This theorem is referenced by:  bd0r  13194  bdth  13200  bdnth  13203  bdnthALT  13204  bdph  13219  bdsbc  13227  bdsnss  13242  bdcint  13246  bdeqsuc  13250  bdcriota  13252  bj-axun2  13284